Problem: Solve for $x$ : $ 3|x + 5| - 3 = -5|x + 5| + 6 $
Answer: Add $ {5|x + 5|} $ to both sides: $ \begin{eqnarray} 3|x + 5| - 3 &=& -5|x + 5| + 6 \\ \\ { + 5|x + 5|} && { + 5|x + 5|} \\ \\ 8|x + 5| - 3 &=& 6 \end{eqnarray} $ Add ${3}$ to both sides: $ \begin{eqnarray} 8|x + 5| - 3 &=& 6 \\ \\ { + 3} &=& { + 3} \\ \\ 8|x + 5| &=& 9 \end{eqnarray} $ Divide both sides by ${8}$ $ \dfrac{8|x + 5|} {{8}} = \dfrac{9} {{8}} $ Simplify: $ |x + 5| = \dfrac{9}{8}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 5 = -\dfrac{9}{8} $ or $ x + 5 = \dfrac{9}{8} $ Solve for the solution where $x + 5$ is negative: $ x + 5 = -\dfrac{9}{8} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& -\dfrac{9}{8} \\ \\ {- 5} && {- 5} \\ \\ x &=& -\dfrac{9}{8} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $8$ $ x = - \dfrac{9}{8} {- \dfrac{40}{8}} $ $ x = -\dfrac{49}{8} $ Then calculate the solution where $x + 5$ is positive: $ x + 5 = \dfrac{9}{8} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} x + 5 &=& \dfrac{9}{8} \\ \\ {- 5} && {- 5} \\ \\ x &=& \dfrac{9}{8} - 5 \end{eqnarray} $ Change the ${ - 5}$ to an equivalent fraction with a denominator of $8$ $ x = \dfrac{9}{8} {- \dfrac{40}{8}} $ $ x = -\dfrac{31}{8} $ Thus, the correct answer is $x = -\dfrac{49}{8} $ or $x = -\dfrac{31}{8} $.